Tuesday, May 27, 2014

History of Mathematics: Zero and Brahmagupta

" Zero is both a number and the numerical digit used to represent that number in numerals."

The number zero first appeared in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe) in 628 AD.  Brahmagupta was an Indian mathematician and astronomer who never wrote any proofs for his work so it is unknown how his mathematics were derived.  Most of his works were composed in elliptic verse so it has a poetic ring to them which was common at that time in Indian mathematics. Brahmagupta did a number of important works including algorithms for square roots, the solution of quadratic equations, and the area of a cyclic quadrilateral, known as Brahmagupta's Formula. Brahmagupta established a set of rules for numbers and the modern version includes:
  • The sum of zero and a negative number is negative.
  • The sum of zero and a positive number is positive.
  • The sum of zero and zero is zero.
  • The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
  • A positive or negative number when divided by zero is a fraction with the zero as denominator.
  • Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. 
  • Zero divided by zero is zero.
As you can see, Brahmagupta not only mentions zero but also basic mathematical rules for computing zero as well.  According to Brahmagupta, zero divided by zero is zero which was then accepted for centuries later.  In the 12th Century, almost 500 years after Brahmagupta, Indian mathematician Bhaskara II showed that zero divided by zero should be infinity because 1 can be divided into an infinite number of pieces of size zero.  This answer was considered correct for centuries as well, until the modern view, that any number divided by zero is undefined, was established.  Zero divided by zero is still a controversial topic discussed today by many mathematicians. What do you think the answer should be?

My personal opinion is that zero divided by zero is undefined so I agree with the modern view.  I believe it is undefined because zero divided by zero it is like asking "how many zeros are in zero"; are there 1, 0, infinitely many? Since this question doesn't really make since, I believe zero divided by zero is undefined.


1 comment:

  1. Good so far - entertaining writing & good history. Just needs a bit more to be an exemplar. Maybe your thoughts on your final question, or some more of the history of zero. (Khwarizmi, for example.)